Abstract

Frame theory is the study of topology based on its open set lattice, and it was studied extensively by various authors. In this paper, we study quotients of intuitionistic fuzzy filters of an intuitionistic fuzzy coframe. The quotients of intuitionistic fuzzy filters are shown to be filters of the given intuitionistic fuzzy coframe. It is shown that the collection of all intuitionistic fuzzy filters of a coframe and the collection of all intutionistic fuzzy quotient filters of an intuitionistic fuzzy filter are coframes.

1. Introduction

Frame theory is topology seen through notions of lattice theory; here one takes the lattice of open sets as the basic notion. The concept of Frames has been studied by many mathematicians including Banaschewski, Dowker, and Johnstone. For details one can refer to [1–3]. In 1965 Zadeh [4] introduced the concept of fuzzy sets as the generalization of ordinary subsets. The concept of ideals of a fuzzy subring was studied by Mordeson and Malik in [5] and Prajapati in [6]. In 1983, Atanassov [7] proposed a generalization of the notion of fuzzy set, known as intuitionistic fuzzy sets. In our earlier paper [8, 9], we introduced the concept of intuitionistic fuzzy frames and intuitionistic fuzzy filters of a frame. In this paper, intuitionistic fuzzy filters of a coframe and intuitionistic fuzzy filters of an intuitionistic fuzzy coframe are studied and examined. Subcoframe of the collection of all intuitionistic fuzzy filters of the coframe is obtained.

2. Preliminaries

In this section we will review some fundamental definitions.

Definition 1. Let be any set; then for any arbitrary , and .

Definition 2 (see [1, 2]). A frame is a complete lattice satisfying the infinite distributive law for any and .

Definition 3 (see [1, 2]). A coframe is a complete lattice satisfying the infinite distributive law for any and .

Definition 4 (see [1, 2]). A subset of a coframe closed under arbitrary meets, finite joins and having unit (top) element and zero (bottom) element of the coframe is called a sub coframe of .

Definition 5 (see [1, 2]). For any coframe , a subset of is a filter if and , .

Definition 6 (see [7]). Let be a nonempty set. An intuitionistic fuzzy set (IFS) of is an object of the form , where and define, respectively, the degree of membership and the degree of nonmembership of the element and , for all .

Definition 7 (see [8]). Let be a coframe then an IFS in is called an intuitionistic fuzzy coframe (IFCF) of if it satisfies the following conditions:(i) for all ,(ii) for arbitrary ,(iii) for all , where and are, respectively, the unit and zero element of the coframe .

Definition 13. Let be an IFS. Let where means and for all . Then is called the generated by .

Theorem 14. Let be two ; then is an and .

Proof. Let ; then
Hence . Similarly, we have . Thus, . Also,(F1)(F2)

Similarly, we have for all .

Therefore, , for all

Similarly, we have for all .

Therefore, , for all .(F3) Since , clearly and since , we have .

Thus if , are , then is an .

Now let be any such that , then,
Hence . Thus is the smallest filter of such that .

Hence .

Proposition 15. Let , be two ; then .

Proof. We have, for , and . Hence . Now since and hence and for
Hence . Therefore .

Theorem 16. Given an arbitrary collection and of , then

Proof. We have for ,
Therefore,
Also and for all ; hence and . Therefore,
From (14) and (15) we have .

Remark 17. If we interchange the roles of and in Theorem 16, only one-sided inequality holds.

Theorem 18. The set of all intuitionistic fuzzy filters of the coframe is a coframe.

Proof. is a complete lattice, which is bounded above by where , for all and bounded below by where
Also for arbitrary collection and of by Proposition 15 and Theorem 16. Hence is a coframe.

4. Intuitionistic Fuzzy Filter of an Intuitionistic Fuzzy Coframe

Definition 19. Let be an intuitionistic fuzzy coframe (IFCF) of and an IFS with . Then is called an intuitionistic fuzzy filter of if(i).(ii) for all . for all .(iii) and where is the unit element of .If is an , then we write .

Theorem 20. Let be an IFCF of and an . Then is an .

Proof. Obviously and(i)(ii)(iii) also clearly and where is the unit element of .

Theorem 21. Let be an IFCF of , and let , be two . Then is also an .

Proof. Clearly also(i)(ii)(iii) also and .

Theorem 22. Let be an IFCF of and an with . Then is an if and only if(i); for all ,(ii),(iii) and .

Proof. Suppose conditions (i), (ii), and (iii) holds to prove that is an .Since , one has
Also and for all .Hence,
Therefore is an .Conversely if is an , we have , , and , from Definition 19.Now for all with , and . Hence .

Theorem 23. Let be an IFCF of , and let , be two ; then is an and , .

Proof. (i) We have and from the proof of Theorem 14.(ii) Now by Theorem 22 since , are IFF(), we have and .Hence by Lemma 10.(iii) Also clearly and .Hence is an by Theorem 22.Again and for every .Hence . Similarly .

Definition 24. Let be an IFCF of , and let , be . Then the quotient of by is denoted as and is defined as
The collection of all quotients of any intuitionistic fuzzy filter of is denoted by .

Theorem 25. Let be an IFCF of , and let be . Then is an . Also .

Proof. Let . Suppose , then and are such that and . Now by Theorem 23 is an . Now by Lemma 10 and since
Hence .Now
also
since .Hence
Now
Similarly .Thus
Now
Similarly .Hence
Also
Now from (27), (29), (31), and (32), is an .Also clearly . Since is an , we have by Theorem 22, . Now by Lemma 10 since , we have . Hence and so . Thus .

Theorem 26. Let be an IFCF of , and let , , be . Then the following holds:(1)if ; then and ,(2)if ; then ,(3),(4).

Proof. Let . Let and .If , then . Therefore and hence .So and hence .Similarly it can be shown that . Let . We have . Also implies that .Hence , and so . Also since is an , . Therefore, . We have . So from Theorem 26(2) we have . We have by Theorem 23 is an and .Hence from Theorem 26(1).Let and .If , then . Now since , by Lemma 10.Also by Theorem 22, implies . Hence .Therefore, by Lemma 10, and hence .So . Thus .

Corollary 27. Let be an IFCF of , and let , be . Then(1),(2),(3).

Proof. Since is an by Theorem 20 and , we have by Theorem 26. Since , we have by Theorem 26. By Theorem 26, also . Hence by Theorem 26.

Theorem 28. Let be an IFCF of , and let , , be . Then .

Proof. We have is an such that and by Theorem 23.Hence and by Theorem 26.Hence
Let , and . For every ,
also
Let , ; then , are and , .Now by Theorem 21 is an , and also
Hence . Thus
Now
Also
from (35).Thus
Consequently from (33) and (40).

Theorem 29. Let be an IFCF of , and let the arbitrary collection , , and be ; then for the arbitrary collection and of quotients of ,

Proof. We have
By Theorem 28.Now from Theorem 23, and .Therefore, by Theorem 26, and .Hence
So,
Hence,
by (42).Also for ,
Hence,
Therefore from (45) and (47).

Remark 30. If we interchange the roles of and in Theorem 29, only one-sided inequality holds.